Invariant Yang–Mills connections with Weyl structure

Abstract

Let G be an n -dimensional ( n ≥ 3 ) compact connected semisimple Lie group, and let g be the canonical metric (the Riemannian metric induced by the Killing form of the Lie algebra g of G ). Assume that D is a torsion-free and left invariant connection in the tangent bundle over the Riemannian manifold ( G , g ) such that ( D , g ) is a Weyl structure with respect to a 1-form ω . Then, a necessary and sufficient condition for the connection D to be a Yang–Mills connection is (i) ω = 0 , or (ii) d ω = 0 and ‖ ω ‖ g 2 = 1 ( n − 2 ) . On the other hand, for a torsion-free connection D with Weyl structure ( D , g , ω ) in the tangent bundle over a Riemannian manifold ( M , g ) , we get the fact that the following statements are equivalent: (i) the curvature tensor field R D of D coincides with the curvature tensor field R D ∗ of the conjugate connection D ∗ of D ; (ii) the Ricci tensor field R i c D of D is symmetric; (iii) d ω = 0 . By virtue of these equivalent statements, we get the following result: for a left invariant connection D with Weyl structure ( D , g , ω ) which is torsion-free in the tangent bundle over an n -dimensional ( n ≥ 3 ) compact connected semisimple Lie group with the canonical metric g , assume that D is not the Levi-Civita connection for g and the curvature tensor field R D of D coincides with the curvature tensor field R D ∗ of D ∗ . Then, a necessary and sufficient condition for the connection D to be a Yang–Mills connection is ‖ ω ‖ g 2 = 1 ( n − 2 ) .